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Register a new userGeometric field theory and weak Euler-Lagrange equation for classical relativistic particle-field systems
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A manifestly covariant, or geometric, field theory for relativistic classical particle-field system is developed. The connection between space-time symmetry and energy-momentum conservation laws for the system is established geometrically without splitting the space and time coordinates, i.e., space-time is treated as one identity without choosing a coordinate system. To achieve this goal, we need to overcome two difficulties. The first difficulty arises from the fact that particles and field reside on different manifold. As a result, the geometric Lagrangian density of the system is a function of the 4-potential of electromagnetic fields and also a functional of particles world-lines. The other difficulty associated with the geometric setting is due to the mass-shell condition. The standard Euler-Lagrange (EL) equation for a particle is generalized into the geometric EL equation when the mass-shell condition is imposed. For the particle-field system, the geometric EL equation is further generalized into a weak geometric EL equation for particles. With the EL equation for field and the geometric weak EL equation for particles, symmetries and conservation laws can be established geometrically. A geometric expression for the energy-momentum tensor for particles is derived for the first time, which recovers the non-geometric form in the existing literature for a chosen coordinate system.
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A general field theory for classical particle-field systems is developed. Compared with the standard classical field theory, the distinguish feature of a classical particle-field system is that the particles and fields reside on different manifolds. The fields are defined on the 4D space-time, whereas each particles trajectory is defined on the 1D time-axis. As a consequence, the standard Noethers procedure for deriving local conservation laws in space-time from symmetries is not applicable without modification. To overcome this difficulty, a weak Euler-Lagrange equation for particles is developed on the 4D space-time, which plays a pivotal role in establishing the connections between symmetries and local conservation laws in space-time. Especially, the non-vanishing Euler derivative in the weak Euler-Lagrangian equation generates a new current in the conservation laws. Several examples from plasma physics are studied as special cases of the general field theory. In particular, the relations between the rotational symmetry and angular momentum conservation for the Klimontovich-Poisson system and the Klimontovich-Darwin system are established.
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